SynthBasics:moogVCF from YampaSynth-0.2

Average Error: 4.8 → 0.8

Time: 10.5s

Precision: binary64

Cost: 33604

Math

\[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]

↓

\[\begin{array}{l} t_1 := \tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\\ \mathbf{if}\;x + \left(y \cdot z\right) \cdot t_1 \leq 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot t_1, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))))

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (tanh (/ t y)) (tanh (/ x y)))))
   (if (<= (+ x (* (* y z) t_1)) 1e+307) (fma (* y t_1) z x) (* z (- t x)))))

C

double code(double x, double y, double z, double t) {
	return x + ((y * z) * (tanh((t / y)) - tanh((x / y))));
}

↓

double code(double x, double y, double z, double t) {
	double t_1 = tanh((t / y)) - tanh((x / y));
	double tmp;
	if ((x + ((y * z) * t_1)) <= 1e+307) {
		tmp = fma((y * t_1), z, x);
	} else {
		tmp = z * (t - x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	return Float64(x + Float64(Float64(y * z) * Float64(tanh(Float64(t / y)) - tanh(Float64(x / y)))))
end

↓

function code(x, y, z, t)
	t_1 = Float64(tanh(Float64(t / y)) - tanh(Float64(x / y)))
	tmp = 0.0
	if (Float64(x + Float64(Float64(y * z) * t_1)) <= 1e+307)
		tmp = fma(Float64(y * t_1), z, x);
	else
		tmp = Float64(z * Float64(t - x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(N[(y * z), $MachinePrecision] * N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Tanh[N[(t / y), $MachinePrecision]], $MachinePrecision] - N[Tanh[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x + N[(N[(y * z), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 1e+307], N[(N[(y * t$95$1), $MachinePrecision] * z + x), $MachinePrecision], N[(z * N[(t - x), $MachinePrecision]), $MachinePrecision]]]

Target

Original	4.8
Target	2.0
Herbie	0.8

\[x + y \cdot \left(z \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right)\right) \]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))))) < 9.99999999999999986e306

   1. Initial program 2.2

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]

   2. Applied egg-rr 0.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \cdot y, z, x\right)} \]

   if 9.99999999999999986e306 < (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))

   1. Initial program 62.2

      \[x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \]

   2. Simplified 62.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot z, \tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right), x\right)} \]
      Proof
      (fma.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y))) x): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))) x)): 1 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 x (*.f64 (*.f64 y z) (-.f64 (tanh.f64 (/.f64 t y)) (tanh.f64 (/.f64 x y)))))): 0 points increase in error, 0 points decrease in error

   3. Taylor expanded in y around inf 5.2

      \[\leadsto \color{blue}{\left(t - x\right) \cdot z + x} \]

   4. Taylor expanded in z around inf 6.6

      \[\leadsto \color{blue}{\left(t - x\right) \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y \cdot z\right) \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right) \leq 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(\tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\right), z, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	1.4
Cost	27332

\[\begin{array}{l} t_1 := \tanh \left(\frac{t}{y}\right) - \tanh \left(\frac{x}{y}\right)\\ \mathbf{if}\;x + \left(y \cdot z\right) \cdot t_1 \leq 10^{+307}:\\ \;\;\;\;x + y \cdot \left(z \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \end{array} \]

Alternative 2
Error	9.9
Cost	7240

\[\begin{array}{l} \mathbf{if}\;y \leq -5.3 \cdot 10^{+166}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+167}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \tanh \left(\frac{t}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \]

Alternative 3
Error	14.7
Cost	6984

\[\begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+21}:\\ \;\;\;\;x + z \cdot \left(t - x\right)\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{+65}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, z, x\right)\\ \end{array} \]

Alternative 4
Error	22.6
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -2.95 \cdot 10^{-239}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-222}:\\ \;\;\;\;z \cdot t\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{-186}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-144}:\\ \;\;\;\;z \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	14.7
Cost	712

\[\begin{array}{l} t_1 := x + z \cdot \left(t - x\right)\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+66}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	17.7
Cost	584

\[\begin{array}{l} t_1 := x + z \cdot t\\ \mathbf{if}\;y \leq -3 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+67}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	22.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;x \leq -1.76 \cdot 10^{-242}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-221}:\\ \;\;\;\;z \cdot t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	22.6
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022325 
(FPCore (x y z t)
  :name "SynthBasics:moogVCF from YampaSynth-0.2"
  :precision binary64

  :herbie-target
  (+ x (* y (* z (- (tanh (/ t y)) (tanh (/ x y))))))

  (+ x (* (* y z) (- (tanh (/ t y)) (tanh (/ x y))))))